Since if the option's price is lower than its intrinsic value (eg. strike price - current stock price for puts), then an arbitrage opportunity arises from buying the option at bargain and then exercising it...

Consequently, an option's price will always be greater than or equal to its intrinsic value

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"Since if the option's price is lower than its intrinsic value (eg. strike price - current stock price for puts), then an arbitrage opportunity arises from buying the option at bargain and then exercising it..."

An options price cannot be lower than its intrinsic value (for any discernab,le amount of time - assuming markets remain open and transactions are occuring), for the simple reason that it will represent "free money" - it will be arbitraged away immediately, as you quite rightly noted in your question.

In kamikaze_pilot's defense, the question is not that naive or simple.

First of all, you need to define what options you are talking about. Consider a digital option for example (which is really fairly vanilla since you can proxy it as a combination of two European calls), which pays 1 of the stock is beyond a certain level at maturity and nothing otherwise. The intrinsic value of the option is just 1 or 0, but would you pay more than 1 to buy such an option ? (what's the point of paying more than whatever you would get at most as a payoff....)

In fact, there is a general point here, a option with a limited upside can have a price lower than the intrinsic value. And this may even apply to European vanilla options, although in very specific circumstances: take a European put option with 0.5y to maturity, 20% implied vol, 100 strike and 5% risk-free rate (ignore all dividends for this example). Now for various spot level, calculate the intrinsic value ( Max(0,strike - spot)) and compare to the option price which you can calculate using Black-Scholes:

Of course, this only happens for deep in-the-money put options. Also it can not happen for American options (there would definitely be an arbitrage here, since you could by the option and exercise it right away).

In terms of arbitrage, consider the following: At time t = 0, your underlying spot is S(0). You buy the stock and a European put option with maturity T and strike K. At time T (maturity), the stock is worth S(T) and your portfolio is either

1) (K - S(T)) + S(T) = K, if S(T) < K

2) or S(T), if S(T) > K

If risk-free rate is r, the accrued cost of the portfolio you set up at time 0 is (P(0) + S(t)) * exp(rT).
To avoid arbitrage, you need this cost to be greater than the minimum of 1) and 2) above, which is K.
You get (P(0) + S(t)) * exp(rT) > K --> P(0) > K*exp(-rT) - S(0). This is the actual arbitrage condition, which is close to saying the intrinsic value is a lower bound of the put option but not quite.